(cf. [1]})
If \(\beta \colon  V \times V \rightarrow {\mathbb {R}}\)  is a symmetric bilinear form,
then \(\Phi _\beta \colon  V \rightarrow V^{\prime }, v \mapsto i_v\beta \)  also is a continuous
linear map, and
\(B := \Phi _\omega ^{-1} \circ \Phi _\beta \colon  V \rightarrow V\)  is continuous
linear with
\( \omega (Bv,v) = \Phi _\omega (Bv)(v)= \Phi _\beta (v)(v) = \beta (v,v), \) 
so that every symmetric bilinear form can be represented by an
element of \(\mathfrak {sp}(V,\omega )\) , and we obtain a topological isomorphism
\(\mathfrak {sp}(V,\omega ) \cong \mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\) . Combining this with
Lemma REF  proves the first assertion.
We now assume that \(V\)  is a real Hilbert space with the scalar product
\((v,w) = \omega (Xv,w)\) , where \(X\)  is as above. Then \(A := X^{-1}\)  is an
invertible skew-symmetric operator with \(\omega (v,w) = (Av,w)\) .
Then the complex linear extension \(A_ to the complex Hilbertspace \) V yields a self-adjoint operator \(iA_.The complex conjugation \)\( of \) V with respect to \(V\) 
now satisfies \(\sigma \circ iA_\sigma = - i A_, sothat \) Spec(iA\( is a compact symmetric subset of \) R\(, notcontaining \) 0\(. We therefore have an orthogonal decomposition\) V V+ V-\( into the positive and negative spectralsubspaces of \) iA. Since \(\sigma (V_\pm ) = V_\mp \) , we obtain
an isomorphism \(V \cong V_+\) , and hence a complex structure
\(I\)  on \(V\) , corresponding to multiplication by \(i\)  on \(V_+\) .
This means that \(v_+ \in V_+\)  with \(v = v_+ + \sigma (v_+)\) 
satisfies
\(\omega (Iv,v)&= (AIv,v)= (AI(v_+ + \sigma (v_+)), v_+ + \sigma (v_+)) \\&= (A_i v_+ - i \sigma (v_+)), v_+ + \sigma (v_+))= i (A_v_+ - \sigma (v_+)), v_+ + \sigma (v_+)) \\&= i \big ((A_v̏_+, v_+) - (v_+, A_v̏_+)\big )= 2 (iA_v̏_+, v_+),\) 

so we obtain a complex structure \(I\)  on \(V\) 
for which the Hamiltonian \(H_I\)  defines a Hilbert structure on
\(V\) . Since \(I\)  is skew-symmetric, \((V,I)\)  inherits the structure
of a complex Hilbert space with respect to the scalar product
\( \langle v, w \rangle := \omega (Iv,w) + i \omega (Iv,Iw) = \omega (Iv,w) + i \omega (v,w). \) 
Therefore our assumption leads to the representation
of \(\omega \)  as \(\omega (v,w) = \mathop {{\rm Im}}\nolimits \langle v, w \rangle \)  for a complex
Hilbert space structure on \(V\) .
\(\Box \)
Involutive Lie algebras with root decomposition
Definition C.1   (a) We call an abelian subalgebra
\({\mathfrak {t}}\)  of the real Lie algebra \({\mathfrak {g}}\)  a compactly embedded
Cartan subalgebra if \({\mathfrak {t}}\)  is maximal abelian and \(\mathop {{\rm ad}}\nolimits {\mathfrak {t}}\) 
is simultaneously diagonalizable on the complexification \({\mathfrak {g}}_with purely imaginary eigenvalues.Then we have a {\it root decomposition}\)\( {\mathfrak {g}}_ {\mathfrak {t}}_ \sum _{\alpha \in \Delta } {\mathfrak {g}}_\alpha , \)\(where \) g= { x g (h t [h,x]= (h)
x}\( and\)\(\Delta := \lbrace  \alpha \in {\mathfrak {t}}_* \setminus \lbrace 0\rbrace  \colon  {\mathfrak {g}}_\alpha \ne  \lbrace 0\rbrace \rbrace \)\( is the corresponding {\it root system}.\) If \(\sigma \colon  {\mathfrak {g}}_{\mathfrak {g}}_ denotes the complex conjugation withrespect to \) g\(, we write \) x* := -(x)\( for \) x g, so that
\({\mathfrak {g}}= \lbrace  x \in {\mathfrak {g}}_ x^* = -x\rbrace \) .
We then have
(I1)
\(\alpha (x) \in {\mathbb {R}}\)  for \(x \in i {\mathfrak {t}}\) .
(I2)
\(\sigma ({\mathfrak {g}}_\alpha ) = {\mathfrak {g}}_{-\alpha }\)  for \(\alpha \in \Delta .\)


Lemma C.2 
 For \(0 \ne  x_{\alpha } \in {\mathfrak {g}}_{\alpha }\)  the subalgebra
\({\mathfrak {g}}_x_\alpha ) := \mathop {{\rm span}}\nolimits \lbrace x_\alpha , x_{\alpha }^*,[x_\alpha , x_{\alpha }^*]\rbrace \) 
is \(\sigma \) -invariant and of one of the following types:
(A)
The abelian type: \([x_\alpha , x_{\alpha }^*] = 0\) , i.e.,
\({\mathfrak {g}}_x_\alpha )\)  is two dimensional abelian.
(N)
The nilpotent type: \([x_\alpha , x_{\alpha }^*] \ne  0\) 
and \(\alpha ([x_\alpha , x_{\alpha }^*]) = 0\) , i.e.,
\({\mathfrak {g}}_x_\alpha )\)  is a three dimensional Heisenberg algebra.
(S)
The simple type: \(\alpha ([x_\alpha , x_{\alpha }^*]) \ne  0\) ,
i.e., \({\mathfrak {g}}_x_\alpha ) \cong \mathop {{\mathfrak {sl} }}\nolimits _2(\) . In this case we distinguish the
two cases:
(CS)
\(\alpha ([x_\alpha , x_{\alpha }^*]) > 0\) , i.e.,
\({\mathfrak {g}}_x_\alpha ) \cap {\mathfrak {g}}\cong \mathop {{\mathfrak {su} }}\nolimits _2(\) , and
(NS)
\(\alpha ([x_\alpha , x_{\alpha }^*]) < 0\) , i.e.,
\({\mathfrak {g}}_x_\alpha ) \cap {\mathfrak {g}}\cong \mathop {{\mathfrak {su} }}\nolimits _{1,1}( \cong \mathop {{\mathfrak {sl} }}\nolimits _2({\mathbb {R}})\) .


Proof.   
First we note that, in view of \(x_\alpha ^* \in {\mathfrak {g}}_{-\alpha }\) ,
[2]} applies, and we see that \({\mathfrak {g}}_x_\alpha )\)  is of one of
the three types (A), (N) and (S). We note that
\(\alpha ([x_\alpha , x_\alpha ^*]) \in {\mathbb {R}}\) 
because of (I2) and \([x_\alpha , x_\alpha ^*] \in i{\mathfrak {t}}\) .
Now it is easy to check that
\({\mathfrak {g}}_x_\alpha ) \cap {\mathfrak {g}}\)  is of type (CS), resp., (NS), according to the sign
of this number.
\(\Box \)
The following proposition provides useful information for the
analysis of invariant cones and orbit projections.
Here we write \(p_{\mathfrak {t}}\colon  {\mathfrak {g}}\rightarrow {\mathfrak {t}}\)  for the
projection along \([{\mathfrak {t}},{\mathfrak {g}}] = \sum _{\alpha } ({\mathfrak {g}}_\alpha + {\mathfrak {g}}_{-\alpha }) \cap {\mathfrak {g}}\) ,
and \(p_{{\mathfrak {t}}^*} \colon  {\mathfrak {g}}^* \rightarrow {\mathfrak {t}}^*\)  for the restriction map.
Proposition C.3 

Let \(x \in {\mathfrak {t}}\) , \(x_\alpha \in {\mathfrak {g}}_\alpha \)  and
\(\lambda \in {\mathfrak {t}}^*\) . Then the following assertions hold:
(i)
\(p_{\mathfrak {t}}(e^{{\mathbb {R}}\mathop {{\rm ad}}\nolimits (x_\alpha - x_\alpha ^*)}x)= x +{\left\lbrace \begin{array}{ll}{\mathbb {R}}^+ \alpha (x) [x_\alpha ^*, x_\alpha ] & \text{ for }\ \alpha ([x_\alpha , x_\alpha ^*]) \le 0 \\[0,2] { \frac{\alpha (x)}{\alpha ([x_\alpha , x_\alpha ^*])}}[x_\alpha ^*, x_\alpha ] & \text{ for }\ \alpha ([x_\alpha , x_\alpha ^*]) > 0.\end{array}\right.}\)
(ii)
\(p_{{\mathfrak {t}}^*}(e^{{\mathbb {R}}\mathop {{\rm ad}}\nolimits ^* (x_\alpha - x_\alpha ^*)}\lambda )= \lambda +{\left\lbrace \begin{array}{ll}{\mathbb {R}}^+ \lambda ([x_\alpha ^*, x_\alpha ]) \alpha & \text{ for }\ \alpha ([x_\alpha , x_\alpha ^*]) \le 0 \\[0,2] \frac{\lambda ([x_\alpha ^*, x_\alpha ])}{\alpha ([x_\alpha , x_\alpha ^*])}\alpha & \text{ for }\ \alpha ([x_\alpha , x_\alpha ^*]) > 0.\end{array}\right.}\)


Proof.   
(i) is an immediate consequence of [3]},
and (ii) follows from (i) and the relation
\(p_{{\mathfrak {t}}^*}(\lambda )(e^{\mathop {{\rm ad}}\nolimits y}x)= \lambda (p_{\mathfrak {t}}(e^{\mathop {{\rm ad}}\nolimits y}x)).\) 
\(\Box \)
Some facts on Fock spaces
Let \(\mathcal {H}\)  be a complex Hilbert space. We endow the \(n\) -fold tensor
product with the canonical Hilbert structure defined by
\( \langle v_1 \otimes \cdots \otimes v_n, w_1 \otimes \cdots \otimes w_n \rangle := \prod _{j = 1}^n \langle v_j, w_j \rangle , \) 
and form the Hilbert space direct sum
\(T(\mathcal {H}) := \widehat{\bigoplus }_{n \in {\mathbb {N}}_0} \mathcal {H}^{\otimes n}\) .
In \(\mathcal {H}^{\otimes n}\)  we write \(S^n(\mathcal {H})\)  for the closed subspace
generated by the symmetric tensors and
\(\Lambda ^n(\mathcal {H})\)  for the closed subspace generated by the
alternating tensors. We thus obtain subspaces
\( S(\mathcal {H}) := \widehat{\bigoplus }_{n \in {\mathbb {N}}_0} S^n(\mathcal {H})\quad \mbox{ and } \quad \Lambda (\mathcal {H}) := \widehat{\bigoplus }_{n \in {\mathbb {N}}_0} \Lambda ^n(\mathcal {H}) \) 
of \(T(\mathcal {H})\)  and write \(P_s\) , resp., \(P_a\)  for the corresponding
orthogonal projections:
\( P_s(f_1 \otimes \cdots \otimes f_n)= \frac{1}{n!} \sum _{\sigma \in S_n} f_{\sigma (1)} \otimes \cdots \otimes f_{\sigma (n)} \) 
and
\( P_a(f_1 \otimes \cdots \otimes f_n)= \frac{1}{n!} \sum _{\sigma \in S_n} \mathop {{\rm sgn}}\nolimits (\sigma )f_{\sigma (1)} \otimes \cdots \otimes f_{\sigma (n)}. \) 
The dense subspace \(S(\mathcal {H})_0 := \sum _{n \ge 0} S^n(\mathcal {H})\)  of \(S(\mathcal {H})\) 
carries an associative
algebra structure, given by
\( f_1 \vee \cdots \vee f_n := \sqrt{n!}P_s(f_1 \otimes \cdots \otimes f_n) \) 
and likewise \(\Lambda (\mathcal {H})_0 := \sum _{n \ge 0} \Lambda ^n(\mathcal {H})\) 
inherits an algebra structure defined by
\( f_1 \wedge \cdots \wedge f_n := \sqrt{n!}P_a(f_1 \otimes \cdots \otimes f_n). \) 
A unit vector \(\Omega \)  in the one-dimensional space
\(S^0(\mathcal {H}) = \Lambda ^0(\mathcal {H})\)  is called a vacuum vector.
Lemma D.1 
 We have
\(T \vee S= \sqrt{{n+m \atopwithdelims ()n}} P_s(T \otimes S) \quad \mbox{ for } \quad T \in S^n(\mathcal {H}), S \in S^m(\mathcal {H})\) 
and
\(T \wedge S= \sqrt{{n+m \atopwithdelims ()n}} P_a(T \otimes S) \quad \mbox{ for } \quad T \in \Lambda ^n(\mathcal {H}), S \in \Lambda ^m(\mathcal {H}).\) 


Proof.   
For the symmetric case we first note that
\(f^n = \sqrt{n!} f^{\otimes n}\) , so that
\( f^n \vee g^m= \sqrt{(n+m)!} P_s(f^{\otimes n} \otimes g^{\otimes m})= \sqrt{{n+m \atopwithdelims ()n}} P_s(f^n \otimes g^m). \) 
Since the elements \(f^n\)  generate \(S^n(\mathcal {H})\)  topologically,
(REF ) follows.
For the alternating case we obtain for
\(T = f_1 \wedge \cdots \wedge f_n\)  and
\(S = g_1 \wedge \cdots \wedge g_m\)  the relation
\(&\ \ \ \ \frac{1}{\sqrt{(n+m)!}}T \wedge S= P_a(f_1 \otimes \cdots \otimes g_m)\\&= P_a(P_a(f_1 \otimes \cdots \otimes f_n) \otimes P_a(g_1 \otimes \cdots \otimes g_m))= \frac{1}{\sqrt{n!m!}} P_a(T \otimes S).\) 

\(\Box \)
Remark D.2  
(a) For the norms of the product of \(T \in S^n(\mathcal {H})\)  and \(S \in S^m(\mathcal {H})\) ,
we obtain with (REF )
\( \frac{1}{\sqrt{(n+m)!}} \Vert T \vee S\Vert = \frac{1}{\sqrt{n!}\sqrt{m!}} \Vert P_s(T \otimes S)\Vert \le \frac{1}{\sqrt{n!}\sqrt{m!}} \Vert T \otimes S\Vert = \frac{\Vert T\Vert }{\sqrt{n!}} \frac{\Vert S\Vert }{\sqrt{m!}}.\) 
From that we derive for \(T \in S^k(\mathcal {H})\)  and \(n \in {\mathbb {N}}\)  the relation
\( \frac{1}{\sqrt{(kn)!}} \Vert T^n\Vert \le \Big (\frac{1}{\sqrt{k!}}\Vert T\Vert \Big )^n, \) 
and for \(T \in S^2(\mathcal {H})\)  we find in particular
\(\Vert e^T\Vert ^2= \sum _{n = 0}^\infty \frac{1}{(n!)^2} \Vert T^n\Vert ^2\le \sum _{n = 0}^\infty \frac{(2n)!}{(n!)^2 2^n} \Vert T\Vert ^{2n},\) 
which converges for \(\Vert T\Vert ^2 < \frac{1}{2}\) .
(b) For \(T \in \Lambda ^n(\mathcal {H})\)  and \(S \in \Lambda ^m(\mathcal {H})\) 
we likewise obtain with (REF )
\( \frac{1}{\sqrt{(n+m)!}} \Vert T \wedge S\Vert \le \frac{\Vert T\Vert }{\sqrt{n!}} \frac{\Vert S\Vert }{\sqrt{m!}},\) 
which leads for \(T \in \Lambda ^2(\mathcal {H})\)  to
\(\Vert e^T\Vert ^2= \sum _{n = 0}^\infty \frac{1}{(n!)^2} \Vert T^n\Vert ^2\le \sum _{n = 0}^\infty \frac{(2n)!}{(n!)^2 2^n} \Vert T\Vert ^{2n}.\) 

Lemma D.3  
Let \(A\)  be an antilinear Hilbert–Schmidt operator on \(\mathcal {H}\) 
and define \(A^*\)  by \(\langle A^*v,w \rangle = \langle Aw,v\rangle \) .
If \(A^* = A\) , then there exists a unique element
\(\widehat{A} \in S^2(\mathcal {H})\)  with
\( \langle \widehat{A}, f_1 \vee f_2 \rangle = \langle Af_1, f_2 \rangle f\quad \mbox{ for } \quad f_1, f_2 \in \mathcal {H}, \) 
and if \(A^* = - A\) , there exists a unique element \(\widehat{A} \in \Lambda ^2(\mathcal {H})\) 
with
\( \langle \widehat{A}, f_1 \wedge f_2 \rangle = \langle Af_1, f_2 \rangle \quad \mbox{ for } \quad f_1, f_2 \in \mathcal {H}. \) 
Moreover, the following assertions hold:
(i)
\(\Vert \widehat{A}\Vert ^2 = {\textstyle {\frac{1}{2}}}\Vert A\Vert _2^2\) .
(ii)
If \(A\)  and \(B\)  are hermitian and antilinear, then
\(\langle \widehat{A}, \widehat{B} \rangle = {\textstyle {\frac{1}{2}}}\mathop {{\rm tr}}\nolimits (AB)\) .
(iii)
If \(A\)  and \(B\)  are skew-hermitian and antilinear, then
\(\langle \widehat{A}, \widehat{B} \rangle = -{\textstyle {\frac{1}{2}}}\mathop {{\rm tr}}\nolimits (AB)\) .


Proof.   
(a) First we consider the case where \(A^* = A\) .
Let \((e_j)_{j \in J}\)  be an orthonormal basis of \(\mathcal {H}\) .
Then we have in \(S^2(\mathcal {H})\)  the relations
\( \Vert e_j^2\Vert ^2 = 2\Vert  e_j \otimes e_j\Vert ^2 = 2\quad \mbox{ and } \quad \Vert e_j \vee e_k\Vert ^2= 2\Vert {\textstyle {\frac{1}{2}}}(e_j \otimes e_k + e_k \otimes e_j)\Vert ^2 = 1, \quad j\ne k. \) 
If \(<\)  denotes a linear order on \(J\) , we thus obtain the
orthonormal basis
\(\frac{1}{\sqrt{2}} e_j^2, \quad e_j \vee e_k, \quad j < k, \quad \mbox{ of } \quad S^2(\mathcal {H}).\) 
This leads to
\(\Vert \widehat{A}\Vert ^2&= \sum _j {\textstyle {\frac{1}{2}}}|\langle \widehat{A}, e_j^2 \rangle |^2+ \sum _{j<k} |\langle \widehat{A}, e_j \vee e_k \rangle |^2 \\&= \sum _j {\textstyle {\frac{1}{2}}}|\langle Ae_j, e_j \rangle |^2+ \sum _{j<k} |\langle Ae_j, e_k \rangle |^2= {\textstyle {\frac{1}{2}}}\Vert A\Vert _2^2.\) 
For \(A^* = -A\)  we similarly get
\(\Vert \widehat{A}\Vert ^2&= \sum _{j<k} |\langle \widehat{A}, e_j \vee e_k \rangle |^2= \sum _{j<k} |\langle Ae_j, e_k \rangle |^2 = {\textstyle {\frac{1}{2}}}\Vert A\Vert _2^2.\) 
(b) Let \(\mathop {{\rm Herm}}\nolimits _2(\mathcal {H})_a\)  denote the complex subspace
of hermitian antilinear
Hilbert–Schmidt operators.
Then the prescription \(\langle A, B \rangle := \mathop {{\rm tr}}\nolimits (AB) = \mathop {{\rm tr}}\nolimits (AB^*)\)  defines a
sesquilinear form on this space with
\( \overline{\mathop {{\rm tr}}\nolimits (AB)} = \mathop {{\rm tr}}\nolimits ((AB)^*) = \mathop {{\rm tr}}\nolimits (B^*A^*) = \mathop {{\rm tr}}\nolimits (BA), \) 
so that it is hermitian. Its restriction to the diagonal
satisfies
\(\langle A,A \rangle = \mathop {{\rm tr}}\nolimits (A^2)= \sum _{j \in J} \langle A^2e_j, e_j \rangle = \sum _{j \in J} \langle Ae_j, Ae_j \rangle = \Vert A\Vert _2^2. \) 
In view of (a), polarization implies that
\(\mathop {{\rm tr}}\nolimits (AB) = \langle A, B \rangle = 2 \langle \widehat{A}, \widehat{B}\rangle \) 
for \(A, B \in \mathop {{\rm Herm}}\nolimits _2(\mathcal {H})_a\) .
(c) For the space \(\mathop {{\rm Aherm}}\nolimits _2(\mathcal {H})_a\)  of skew-hermitian antilinear
Hilbert–Schmidt operators the same argument works with
\(\langle A, B \rangle := -\mathop {{\rm tr}}\nolimits (AB) = \mathop {{\rm tr}}\nolimits (AB^*)\) .
\(\Box \)
